When does $e^{f(x)}$ have an antiderivative?

Question

today I tried to integrate $x^x$ by applying a reverse chain rule which turned out to be false. I was told $\int e^{f(x)}\,dx$ can be done when $f(x)$ is linear. This made me wonder what conditions we can find so that $\int e^{f(x)}\,dx$ can be expressed in terms of elementary functions, but I'm not sure what to do.

Answer 1

When $f$ is a polynomial, it is a consequence of a celebrated theorem by Liouville that $e^{f(x)}$ has an elementary antiderivative iff there is a polynomial $h$ such that $1=h'+hf$. This implies that $f$ has degree at most 1. In particular, the function $e^{x^2}$ does not have an elementary antiderivative.

See How can you prove that a function has no closed form integral?.